Solve the equation: e^2x - 5e^x + 4 = 0

Question

Solve the equation. Give exact answers.
e^2x − 5e^x + 4 = 0

I have no idea how to factor something with e's in it. I don't know if factoring is the way you go about an equation with e's in it. Please explain the answer's steps.

 

Answer 1

Apply the rules of exponents on the first term.

 

(e^x)² - 5e^x + 4 = 0

 

Let       q = e^x

 

 

q² - 5q + 4 = 0

 

 

Now we have a quadratic equation.

 

Solve for q using the FOIL method.

 

(q -  4)(q - 1) = 0

 

q = 4       ,         q = 1

 

 

Substitute these values of q to find x.

 

e^x = 4              and            e^x = 1

 

 

Use logarithms to solve for x.